# On Numbers And Games

John Conway

## Preface

This book was written to bring to light a relation between two of its author's favourite subjects-the theories of transfinite numbers and mathematical games. A few connections between these have been known for some time, but it appears to be a new observation that we obtain a theory at once simpler and more extensive than Dedekind's theory of the real numbers just by defining numbers as the strengths of positions in certain games. When we do this the usual properties of order and the arithmetic operations follow almost immediately from definitions that are naturally suggested, so that it was quite an amusing exercise to write the zeroth part of the book as if these definitions had arisen instead from an attempt to generalise Dedekind's construction!

However, we suspect that there will be many readers who are more interested in playing games than philosophising about numbers. For these readers we offer the following words of advice. Start reading Chapter 7, on playing several games at once, and find an interested friend with whom to play a few games of the domino game described there. In this it's easy to see why and give Left one and two moves advantage respectively - when you feel you vaguely understand why gives him just half of a move's advantage, you might like to read Chapter 0, which explains how the simplest numbers arise. You should then find no difficulty in reading the rest of the book without knowing any more about numbers than that "ordinals" are a certain kind of (usually infinite) whole number, and that the Author has strange idiosyncracies which make him use capital letters for certain very large infinite collections.

Many friends have helped me to write this book, often without being aware of the fact. I owe an especial debt to Elwyn Berlekamp and Richard Guy, with whom I am currently preparing a more extended book on mathematical games which should overlap this one in several places. The book would never have appeared without the repeated gentle proddings that came from Anthony Watkinson of Academic Press; it would have contained many errors were it not for the careful reading of Paul Cohn as editor, and the quality of the printing and layout could never have been so high without the detailed attentions of Ron Hitchings and the staff of the printers at Page Bros of Norwich. Others whose comments have affected more than one page are Mike Christie, Aviezri Fraenkel, Mike Guy, Peter Johnstone, Donald Knuth and Simon Norton. The varied nature of the advice they gave is neatly encapsulated in the following lines from Bunyan's *Apology for his Book (Pilgrim's Progress):*

*Some said 'John, print it'; others said, 'Not so.'
Some said 'It might do good'; others said, 'No.'*

October 1975

J.H.C.

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